Hypergeometric function \(\,_1F_1\) and related

Text explaining this chapter

• Hypergeometric Functions \(\,_1F_1\) (Kummer) and \(U\) (Tricomi)
  • Kummer’s Confluent Hypergeometric Function \({}_1F_1(a,b;x)\)
    • ctx.hyperg_1f1()
  • Regularized Kummer Confluent Hypergeometric Function, \({}_1\widetilde{F}_1(a,b;x)\)
    • ctx.hyperg_1f1r()
  • Tricomi’s Confluent Hypergeometric Function, \(U(a,b;x)\)
    • ctx.hyperg_u()
  • Generalized Laguerre polynomials, \(L^{(a)}_n (x)\)
    • ctx.laguerre_l()
  • Hermite polynomial (physicist), \(H_n(x)\)
    • ctx.hermite_h()
  • Hermite polynomials (probabilist) \(\operatorname{He}_n(x)\)
    • math53.hermite_he()
• Incomplete gamma functions
  • Lower normalized incomplete gamma function, \(P(a,x)\)
    • ctx.gamma_p()
  • Upper normalized incomplete gamma functions , \(Q(a,x)\)
    • ctx.gamma_q()
  • Lower non-normalized incomplete gamma function, \(\gamma(a,x)\)
    • ctx.gamma_lower()
  • Upper non-normalized incomplete gamma function, \(\Gamma(a,x)\)
    • ctx.gamma_upper()
  • Tricomi’s entire incomplete gamma function: \(\gamma^*(a,x)\)
    • ctxflint.gamma_tricomi()
  • Derivative of the incomplete gamma function
    • ctx.gamma_p_prime()
  • Inverse of the real lower normalised incomplete gamma function, \(P^{-1}(a, q)\)
    • ctx.real_gamma_p_inv()
  • Inverse of the real upper normalised incomplete gamma function, \(Q^{-1}(a, q)\)
    • ctx.real_gamma_q_inv()
  • Inverse (on parameter \(a\)) of the real lower normalised incomplete gamma function
    • ctx.real_gamma_p_inva()
  • Inverse (on parameter \(a\)) of the real upper normalised incomplete gamma function
    • ctx.real_gamma_q_inva()
• Coulomb, Whittaker and parabolic cylinder function
  • Regular Coulomb wave function \(F_{\ell}(\eta,x)\)
    • ctx.coulomb_f()
  • Irregular Coulomb wave function \(G_{\ell}(\eta,z)\)
    • ctx.coulomb_g()
  • Irregular Coulomb wave function \(H^{(+)}_{\ell}(\eta,x)\)
    • ctx.coulomb_hplus()
  • Irregular Coulomb wave function \(H^{(-)}_{\ell}(\eta,x)\)
    • ctx.coulomb_hminus()
  • Whittaker function \(M_{\kappa, \mu}(x)\)
    • math53.whittaker_m()
  • Whittaker function \(W_{\kappa, \mu}(x)\)
    • math53.whittaker_w()
  • Parabolic cylinder function \(D_{\nu}(x)\)
    • math53.cylinder_d()
  • Parabolic cylinder function \(U(a, x)\)
    • math53.cylinder_u()
  • Parabolic cylinder function \(V(a,x)\)
    • math53.cylinder_v()
  • Parabolic cylinder function \(W(a,x)\)
    • ctxflint.cylinder_w()
• Error function and related functions
  • Error function, \(\mathrm{erf}(x)\)
    • ctx.erf()
  • Complementary error function, \(\mathrm{erfc}(x)\)
    • ctx.erfc()
  • Inverse of the real error function, \(\mathrm{erf}^{-1}(x)\)
    • ctx.erf_inv()
  • Inverse of the real complementory error function, \(\mathrm{erfc}^{-1}(x)\)
    • ctx.erfc_inv()
  • Standard normal density function \(\phi(x)\)
    • ctx.ndens()
  • Standard normal cumulative distribution function \(\Phi(x)\)
    • ctx.ndis()
  • Imaginary error function, \(\mathrm{erfi}(x)\)
    • ctx.erfi()
  • Dawson integral, \(F(x)\)
    • math53.dawson()
  • Faddeeva function, \(w(z)\)
    • math53.faddeeva()
  • Fresnel sine integral, \(S(x)\)
    • ctx.fresnel_s()
  • Fresnel cosine integral, \(C(x)\)
    • ctx.fresnel_c()
  • Owen’s T function, \(T(h,a)\)
    • ctx.owen_t()
• Exponential integrals, and related functions
  • Exponential integral \(E_1(x)\)
    • ctx.exp_integral_e1()
  • Exponential integral \(\mathrm{Ei}(x)\)
    • ctx.exp_integral_ei()
  • Logarithmic integral \(\mathrm{li}(x)\)
    • ctx.log_integral()
  • Hyperbolic sine integral \(\mathrm{Shi}(x)\)
    • ctx.sinh_integral()
  • Hyperbolic cosine integral \(\mathrm{Chi}(x)\)
    • ctx.cosh_integral()
  • Generalized exponential integral \(E_n(x)\)
    • ctx.exp_integral_en()
  • Sine integral \(\mathrm{Si}(x)\)
    • ctx.sin_integral()
  • Cosine integral \(\mathrm{Ci}(x)\)
    • ctx.cos_integral()